Properties

Label 2-6012-1.1-c1-0-10
Degree $2$
Conductor $6012$
Sign $1$
Analytic cond. $48.0060$
Root an. cond. $6.92864$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.91·5-s + 1.60·7-s + 5.85·11-s + 0.924·13-s − 6.27·17-s − 2.72·19-s − 0.550·23-s + 10.3·25-s − 0.556·29-s + 9.36·31-s − 6.29·35-s + 6.50·37-s − 2.46·41-s − 2.43·43-s − 4.96·47-s − 4.40·49-s + 3.80·53-s − 22.8·55-s + 10.0·59-s − 9.11·61-s − 3.61·65-s − 2.78·67-s − 12.9·71-s + 15.6·73-s + 9.41·77-s + 1.12·79-s − 16.5·83-s + ⋯
L(s)  = 1  − 1.74·5-s + 0.608·7-s + 1.76·11-s + 0.256·13-s − 1.52·17-s − 0.625·19-s − 0.114·23-s + 2.06·25-s − 0.103·29-s + 1.68·31-s − 1.06·35-s + 1.06·37-s − 0.384·41-s − 0.371·43-s − 0.724·47-s − 0.629·49-s + 0.523·53-s − 3.08·55-s + 1.30·59-s − 1.16·61-s − 0.448·65-s − 0.340·67-s − 1.54·71-s + 1.83·73-s + 1.07·77-s + 0.126·79-s − 1.81·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $1$
Analytic conductor: \(48.0060\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.419950045\)
\(L(\frac12)\) \(\approx\) \(1.419950045\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 3.91T + 5T^{2} \)
7 \( 1 - 1.60T + 7T^{2} \)
11 \( 1 - 5.85T + 11T^{2} \)
13 \( 1 - 0.924T + 13T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 + 2.72T + 19T^{2} \)
23 \( 1 + 0.550T + 23T^{2} \)
29 \( 1 + 0.556T + 29T^{2} \)
31 \( 1 - 9.36T + 31T^{2} \)
37 \( 1 - 6.50T + 37T^{2} \)
41 \( 1 + 2.46T + 41T^{2} \)
43 \( 1 + 2.43T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 - 3.80T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 9.11T + 61T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 - 1.12T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 7.96T + 89T^{2} \)
97 \( 1 + 4.68T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.230532882148054433059932314807, −7.39677084973947636849604694004, −6.65291359948800113227687169664, −6.27319972223877666360981627066, −4.82556788884996335169594672030, −4.30724959064858760728687826068, −3.91783720235151907917051842959, −2.94809382749922121961845169926, −1.71943554521531121787282913663, −0.64410373298850437470337297996, 0.64410373298850437470337297996, 1.71943554521531121787282913663, 2.94809382749922121961845169926, 3.91783720235151907917051842959, 4.30724959064858760728687826068, 4.82556788884996335169594672030, 6.27319972223877666360981627066, 6.65291359948800113227687169664, 7.39677084973947636849604694004, 8.230532882148054433059932314807

Graph of the $Z$-function along the critical line